What is responsible for quantization?

[quasar_4]

I'm reviewing my undergraduate quantum mechanics from a while back, and I am not quite sure I understand this correctly. I seem to recall being taught that quantization arises from the imposition of boundary conditions (in the mathematical sense).

But this isn't quite the same as saying that a particle is in a bound state. I know that particles being bound is related to their total energy vs. the potential they're exposed to, but I'm feeling fuzzy on how these are related. Can anyone help to clarify?

Specifically, it's extremely clear to me why energy is quantized in a system like the particle in the box, or even a delta function potential. But it's not intuitively clear to me why energy is quantized for the hydrogen atom - I know that it is, and also that the electron is in a bound state, but I can't *see* this boundedness in the same way that I can for the particle in the box... the mathematics is clear, but I can't think of a great argument in terms of pure physics.

 

[scorpion990]

I was going to make this topic, too.
As I understand it, the energy of the particle may not actually be quantized. The particle can exist in a superposition of states. This state has no definite energy, and our knowledge of it cannot be known exactly because of the Uncertainty Principle.

As I understand it, the particle's observable energy is quantized. It's "actual" energy cannot be determined exactly. But of course... I'm probably wrong.

Would anybody like to recommend a book on the philosophy of quantum mechanics?

 

[inempty]

When a particle is in a bound state, its energy isn't greater than its potential energy at infinity.So at infinity we have an asymptotic exponential-decay solution which corresponds to the boundary condition. On the other hand, an asymptotic exponential-decay boundary condition implies the total energy is less than the potential energy at infinity.

I think it's difficult to find a way to see the boundedness in general, in fact we can't even in the case of an electron in a potential well with finite depth.

 

[bcrowell]

The difference between hydrogen and a particle in an impenetrable box is really just a matter of degree. Start with an impenetrable box. Then lower the potential outside the box to some large but finite value. Then round the corners of the potential. Now you have something like the potential experienced by a nucleon in a nucleus. Now change the shape of the potential a little more, and you have the hydrogen atom. Nothing changes qualitatively while you make these modifications.

One way to see that binding is necessary (although not necessarily sufficient) for energy quantization is that even if you have a potential well with finite depth, there are continuum states. If you toss an electron out into the vacuum in some region far away from the well, there is no way for the electron to "know" that there is some potential well somewhere far away. Nothing will stop you from initializing the state of the electron with any energy you choose.

A good way to look at it that doesn't depend at all on the exact shape of the potential is to imagine the continuum states as living in a very large box with sides of length L, which we'll eventually imagine approaching infinity. Somewhere inside the large box is a potential well of arbitrary shape, which is restricted to a region of small dimensions S. Without the small well, we would have a dense but finite set of discrete states determined by the boundary conditions of the L-box. When we add the small well, it just perturbs these states, mixing them and producing a new set of states with different energies. Among these are a few states that are basically tricky Fourier superpositions of the L-box's sine-wave states that are highly localized in the region of the S-box. These states have lowered energies.

 

[inempty]

In fact, we can't talk about the position of the electron when it's in an eigenstate of energy.
It's the electron's energy that decides whether its state is a continuum state. And if we want to toss an electron "out of" the potential well, we must give it enough energy.

The way you suggest to see the binding is a heuristic way but I think there is a fatal fallacy. As in the continuum case the difference of energy between two states(zero) before the perturbation is certainly less than the perturbation i.e the potential well confined in the small region, it shouldn't be regarded as perturbation at all. I think we could assert that it's the potential well,not a much larger box(I can't imagine that the box really exists!) that produce quantization.

 

[Demystifier]

There is a simple intuitive way to understand energy quantization in an arbitrary potential, explained in the QM textbook by Eisberg and Resnick. Let me sketch the idea.

Take ANY energy smaller than the potential at infinity. To find the wave function, you need to solve a differential equation (time-independent Schrödinger equation). A solution exists for ANY energy. However, for most energies the wave function diverges at infinity. Such wave functions are not physically acceptable, so the corresponding energies are not allowed. The energy must be fine tuned in order to have a solution that does not diverge. The simplest way to understand the origin of divergence is to consider a 1-dimensional case and think about solving the differential equation numerically, by starting from an initial point at which the wave function is freely chosen. The details are left as an exercise for the readers. (See also the book above.)

 

[heldervelez]

Quantization results of electronic orbits beeing resonances - standing waves, limited by 'c', geometry and energy, give a limited set of possible states.
Energy is associated to frequency and half-wavelengths determine the possible solutions.
How can energy be localized, bounded, without coherence, resonance?

 

[Frame Dragger]

Quantization results of electronic orbits beeing resonances - standing waves, limited by 'c', geometry and energy, give a limited set of possible states.
Energy is associated to frequency and half-wavelengths determine the possible solutions.
How can energy be localized, bounded, without coherence, resonance?

If that last sentence is a trick question, the answer would be, "Past the event horizon of a Black Hole it's possible, along with all of the lost socks and green slime." If it isn't I think Demystifier's answer was more helpful to someone asking about the basic reason why energy is quantized.

 

[PhilDSP]

I'm reviewing my undergraduate quantum mechanics from a while back, and I am not quite sure I understand this correctly. I seem to recall being taught that quantization arises from the imposition of boundary conditions (in the mathematical sense).

A mathematician uses the term "boundary conditions" in a way analogous to the way a physicist establishes a gauge. Typically a solution to differential equation will have "floating" numerical values that need to be anchored to some initial or "boundary" conditions.

But if the author of the text wasn't so much of a mathematician, he or she may have meant what you may have thought about modeled spatial or temporal boundaries which cause a reflection or compression or refraction of energy. That would seem to make the most sense based on what you wrote.